The former USA and Egypt coach believes his son, who eventually joined Toronto FC from Roma in January, was overlooked in favour of Kim Kallstrom by the Gunners
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Bradley returned to MLS to join Canadian franchise Toronto FC in a high-profile switch during the January transfer window, ending his two-year spell with Serie A giants Roma.
The 26-year-old was on the radar of Arsenal, before manager Arsene Wenger opted to sign Sweden midfielder Kim Kallstrom on a short-term loan deal.
And Bob, who coaches Norwegian outfit Stabaek after spells in charge of the US and Egyptian national teams, believes his son's nationality cost him a move to the Premier League.
"I think American players and coaches have to fight really hard for respect," Bob said in an interview on Slate.com's 'Hang Up and Listen' podcast.
"In January, Arsenal (were) looking to add a midfielder, and they chose Kim Kallstrom.
"Kim Kallstrom's not a bad player, but I think Michael feels pretty strongly that he's better, and so Arsene Wenger must not feel that way, and Ivan Gazidis must not feel that way.
"So sometimes, no matter what you do, you don’t get the respect you think you deserve."